Linear Programming

Step 1: Formulate the problem.
We need to maximize under the constraints: - , - , - , - , - .

Step 2: Graph the constraints.
Graph the inequalities on a coordinate plane: - is a line with slope , - is a line with slope , - The region of feasible points is where all inequalities are satisfied, which is the region on or below these lines.

Step 3: Identify the corner points.
The corner points of the feasible region are the points of intersection of the lines: - intersects at , - The intercepts for the lines give additional points at and .

Step 4: Calculate the value of at each corner point.
At , , At , , At , .

Step 5: Conclusion.
The maximum value of is , which occurs at the point .

Step 1: Formulate the problem.
We need to minimize under the constraints: - , - , - , - .

Step 2: Graph the constraints.
Graph the inequalities on a coordinate plane: - is a line with slope , - is a line with slope , - The region of feasible points is where all inequalities are satisfied, which is the region that lies on or above these lines.

Step 3: Identify the corner points.
The corner points of the feasible region are the points of intersection of the lines: - intersects at , - The intercepts for the lines give additional points at and .

Step 4: Calculate the value of at each corner point.
At , , At , , At , .

Step 5: Conclusion.
The minimum value of is , which occurs at the point .